Hermitian positive definite matrix

In summary, the conversation was about proving the inequality x*Px < or eq. x*Qx for all x in C^n, using the definition of a Hermitian positive definite matrix. It was suggested to consider the special case where P and Q are diagonal, and then use the spectral theorem to generalize the proof for all Hermitian positive definite matrices.
  • #1
math8
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Let P and Q be Hermitian positive definite matrices.
We prove that x*Px < or eq. x*Qx, for all x in C^n (C : complex numbers) if and only if x*Q^-1 x < or eq. x*P^-1 x for all x in C^n.

I guess I should use the definition of a hermitian positive definite matrix being
x*Px > 0 , for all x in C^n but I am not sure how to proceed to get both the P and Q in the inequality.

Should I try and multiply both sides of the inequality by x and x*?
 
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  • #2
Try proving it in the special case where P and Q are also diagonal. Then for the general case, use the fact that you can write

[tex]P = A^{-1}D_P A[/tex]
[tex]Q = B^{-1}D_Q B[/tex]

where [tex]D_P[/tex] and [tex]D_Q[/tex] are both diagonal. (This is the spectral theorem.)
 

1. What is a Hermitian positive definite matrix?

A Hermitian positive definite matrix is a square matrix that is equal to its own conjugate transpose, and has all positive eigenvalues. In other words, it is a symmetric matrix with all positive eigenvalues.

2. What is the significance of a Hermitian positive definite matrix?

Hermitian positive definite matrices have many important applications in mathematics and science. They are used in optimization problems, statistics, and quantum mechanics. Additionally, they have efficient computational properties, making them useful in numerical analysis.

3. How can I determine if a matrix is Hermitian positive definite?

A matrix can be determined to be Hermitian positive definite by checking its eigenvalues. If all of the eigenvalues are positive, then the matrix is Hermitian positive definite. Additionally, the matrix must be symmetric, meaning that it is equal to its own transpose.

4. Can a non-square matrix be Hermitian positive definite?

No, a non-square matrix cannot be Hermitian positive definite. The definition of a Hermitian positive definite matrix requires that it must be a square matrix. However, non-square matrices can have similar properties, such as being positive definite or Hermitian.

5. How is a Hermitian positive definite matrix used in quantum mechanics?

In quantum mechanics, Hermitian positive definite matrices are used to represent physical observables, such as energy and momentum. The eigenvalues of these matrices correspond to the possible values of the observable, and the eigenvectors correspond to the states in which those values are measured.

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